Counting Hopf-Galois Structures on Cyclic Field Extensions of Squarefree Degree
aa r X i v : . [ m a t h . R A ] S e p COUNTING HOPF-GALOIS STRUCTURES ONCYCLIC FIELD EXTENSIONS OF SQUAREFREEDEGREE
ALI A. ALABDALI AND NIGEL P. BYOTT
Abstract.
We investigate Hopf-Galois structures on a cyclic fieldextension
L/K of squarefree degree n . By a result of Greither andPareigis, each such Hopf-Galois structure corresponds to a groupof order n , whose isomorphism class we call the type of the Hopf-Galois structure. We show that every group of order n can occur,and we determine the number of Hopf-Galois structures of eachtype. We then express the total number of Hopf-Galois structureson L/K as a sum over factorisations of n into three parts. Asexamples, we give closed expressions for the number of Hopf-Galoisstructures on a cyclic extension whose degree is a product of threedistinct primes. (There are several cases, depending on congruenceconditions between the primes.) We also consider one case wherethe degree is a product of four primes. Introduction and Statement of Results
Let
L/K be a finite Galois extension of fields with Galois group Γ.Then the group algebra K [Γ] is a K -Hopf algebra, and its action on L endows L/K with a Hopf-Galois structure. In general, this is oneamong many possible Hopf-Galois structures on
L/K . Greither andPareigis [GP87] showed that these Hopf-Galois structures correspondto certain regular subgroups G in the group Perm(Γ) of permutationsof the underlying set of Γ. Finding all Hopf-Galois structures in anyparticular case then becomes a combinatorial question in group theory.The groups Γ and G necessarily have the same order, but need not beisomorphic. We refer to the isomorphism type of G as the type of thecorresponding Hopf-Galois structure. Date : November 12, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Hopf-Galois structures; field extensions; groups ofsquarefree order.The first-named author acknowledges support from The Higher Committee forEducation Development in Iraq.
There is a substantial literature on Hopf-Galois structures on variousclasses of field extension. We mention a few results now, and someothers in the final section of this paper. Let p be an odd prime. Acyclic field extension of degree p m admits precisely p m − Hopf-Galoisstructures, all of cyclic type [Koh98]. An elementary abelian extensionof degree p m admits many more: there are at least p m ( m − − ( p − p > m [Chi05],and there are also some of nonabelian type if m ≥ G of anyHopf-Galois structure must be soluble [Byo15], although for a soluble,nonabelian Galois group Γ there can be Hopf-Galois structures whosetype is not soluble. Recently, Crespo, Rio and Vela [CRV16] haveinvestigated those Hopf-Galois structures on an extension L/K whicharise by combining Hopf-Galois structures on
L/F and on
F/K forsome intermediate field F .In this paper, we investigate Hopf-Galois structures on cyclic ex-tensions L/K of arbitrary squarefree degree. Thus we consider cyclicextensions whose degree has a prime factorisation at the other extremeto those treated in [Koh98]. We intend to discuss Hopf-Galois struc-tures on arbitrary Galois extensions of squarefree degree in a futurepaper.The type of a Hopf-Galois structure on a cyclic extension of square-free degree n could potentially be any group G of order n . There maybe many of these. Indeed, H¨older [H¨ol95] showed that the number ofisomorphism types of groups of squarefree order n is given by(1) X de = n Y p | d (cid:18) p v ( p,e ) − p − (cid:19) , where the sum is over ordered pairs ( d, e ) of positive integers such that de = n , the product is over primes p dividing d , and v ( p, e ) is thenumber of distinct prime factors q of e with q ≡ p ). It is clearthat, as n varies over all squarefree integers, the expression (1) canbecome arbitrarily large.It is an immediate consequence of Theorem 1 below that, for eachgroup G of order n , the number of Hopf-Galois structures of type G ona cyclic extension of degree n cannot be zero. Thus all possible typesdo in fact occur. The cyclic extensions of squarefree degree thereforeform a class for which both the number of distinct types of Hopf-Galoisstructures on a given extension, and the number of distinct prime fac-tors of the degree of a given extension, may be arbitrarily large. To thebest of our knowledge, this is the first class of extensions with these OUNTING HOPF-GALOIS STRUCTURES 3 properties for which it has been possible to enumerate all Hopf-Galoisstructures. For comparison, we mention that, when the Galois group Γis a nonabelian simple group, the number of prime factors of | Γ | may bearbitrarily large, but there are only two Hopf-Galois structures, bothof type Γ [Byo04a]. On the other hand, for Galois extensions of de-gree p p p , where p , p , p are distinct odd primes satisfying certaincongruence conditions, Kohl [Koh16] has determined all Hopf-Galoisstructures for each possible Galois group. In this case, the numberof distinct types may be arbitrarily large, but the number of primesdividing the degree is of course fixed at three.We will see in Proposition 3.5 that each group G of squarefree order n gives rise to a factorisation n = dgz of n , in which g (respectively, z )is the order of the commutator subgroup G ′ (respectively, the centre Z ( G )) of G . We can now state the first of our two main results. Theorem 1.
Let
L/K be a cyclic extension of fields of squarefree degree n , and let G be any group of order n . Let z = | Z ( G ) | , g = | G ′ | and d = n/ ( gz ) . Then L/K admits precisely ω ( g ) ϕ ( d ) Hopf-Galoisstructures of type G , where ϕ is Euler’s totient function and ω ( g ) isthe number of (distinct) prime factors of g . Our second result gives the total number of Hopf-Galois structures.
Theorem 2.
The number of Hopf-Galois structures on a cyclic fieldextension of squarefree degree n is (2) X dgz = n ω ( g ) µ ( z ) Y p | d (cid:0) p v ( p,g ) − (cid:1) , where the product is over ordered triples ( d, g, z ) of natural numberswith dgz = n . Here µ is the M¨obius function. We remark that (2) has a similar shape to H¨older’s formula (1),although with a sum over factorisations into three parts rather thantwo. In both cases, the term for each factorisation involves a productover primes p dividing d , in which the contribution corresponding to p does not depend on d and p alone. (In (1) it depends on e , and in (2)on g .) 2. Preliminaries on Hopf-Galois structures
Let
L/K be a field extension of finite degree, and let H be a cocom-mutative K -Hopf algebra acting on L . We write ∆ : H → H ⊗ K H and ǫ : H → K for the comultiplication and counit maps on H , anduse Sweedler’s notation ∆( h ) = P ( h ) h (1) ⊗ h (2) . We will say that theaction of H on L makes L into an H -module algebra if h · ( xy ) = ALI A. ALABDALI AND NIGEL P. BYOTT P ( h ) ( h (1) · x ) ⊗ ( h (2) · y ) and h · k = ǫ ( h ) k for all h ∈ H , all x , y ∈ L andall k ∈ K . A Hopf-Galois structure on L consists of a Hopf algebra H acting on L so that L is an H -module algebra and the K -linear map θ : L ⊗ K L → Hom K ( H, L ) is bijective, where θ ( x ⊗ y )( h ) = x ( h · y ) for x , y ∈ L and h ∈ H .When L/K is separable, Greither and Pareigis [GP87] used descenttheory to show how all Hopf-Galois structures on
L/K could be de-scribed in group-theoretic terms. We consider here only the specialcase where
L/K is a Galois extension in the classical sense (that is,
L/K is normal as well as separable). Let Γ = Gal(
L/K ) be the Galoisgroup of
L/K . Then the Hopf-Galois structures on
L/K correspondbijectively to subgroups G of Perm(Γ) which are regular on Γ and arenormalised by the group λ (Γ) of left translations by elements of Γ. Re-call that a group G acting on a set X is regular if the action is transitiveon X and the stabiliser of any point is trivial.The direct determination of all regular subgroups in Perm(Γ) nor-malised by λ (Γ) is often difficult as the group Perm(Γ) is large. How-ever, the condition that λ (Γ) normalises G means that Γ is containedin the holomorph Hol( G ) = G ⋊ Aut( G ) of G , where the latter groupis viewed as a subgroup of Perm(Γ) and is usually much smaller thanPerm(Γ). We may then view Γ as acting on G , and this action is againregular. If the isomorphism types of groups G ∗ of order | Γ | admit amanageable classification, the Hopf-Galois structures on L/K can bedetermined by considering each G ∗ in turn and finding the regular sub-groups Γ ∗ of Hol( G ∗ ) which are isomorphic to Γ. This leads to thefollowing result, cf. [Byo96, Cor. to Prop. 1] or [Chi00, § Lemma 2.1.
Let
L/K be a finite Galois extension of fields with Galoisgroup Γ , and, for any group G with | G | = | Γ | , let e ′ ( G, Γ) be the numberof regular subgroups of Hol( G ) isomorphic to Γ . Then the number e ( G, Γ) of Hopf-Galois structures on L/K of type G is given by e ( G, Γ) = | Aut(Γ) || Aut( G ) | e ′ ( G, Γ) . Moreover, the total number of Hopf-Galois structures on
L/K is givenby P G e ( G, Γ) , where the sum is over all isomorphism types G of groupsof order | Γ | . Preliminaries on groups of squarefree order
We will call a finite group a C -group if all its Sylow subgroups arecyclic. In particular, any group of squarefree order is a C -group. All C -groups are metabelian, so C -groups can in principle be classified OUNTING HOPF-GALOIS STRUCTURES 5 [Rob96, 10.1.10]. This classification is given in a rather explicit form ina paper of Murty and Murty [MM84], who investigated the asymptoticbehaviour of the number of C -groups of order up to a given bound.We state their classification result, in the special case of groups ofsquarefree order, as Lemma 3.2 below. Notation 3.1.
For an integer N ≥
1, we denote by Z N the ring Z /N Z of integers modulo N , and by U ( N ) the group of units in Z N . We writeord N ( a ) for the order of an element a ∈ U ( N ). Abusing notation, wewill often use the same symbol for an element of Z and its class in Z N .We write 1 G for the identity element of a group G . Lemma 3.2.
Let n be squarefree. Then any group of order n has theform G ( d, e, k ) = h σ, τ : σ e = τ d = 1 G , τ στ − = σ k i where n = de , gcd( d, e ) = 1 and ord e ( k ) = d . Conversely, any choiceof d , e and k satisfying these conditions gives a group G ( d, e, k ) of order n . Moreover, two such groups G ( d, e, k ) and G ( d ′ , e ′ , k ′ ) are isomorphicif and only if d = d ′ , e = e ′ , and k , k ′ generate the same cyclic subgroupof U ( e ) .Proof. This follows from [MM84, Lemmas 3.5 & 3.6]. (cid:3)
Remark 3.3.
The existence of k with ord e ( k ) = d implies that d divides ϕ ( e ) = | U ( e ) | . Thus there may be many factorisations n = de of n for which no groups G ( d, e, k ) occur. Remark 3.4.
We note in passing how H¨older’s formula (1) followsfrom Lemma 3.2. For fixed d and e , the number of isomorphism types ofgroup G ( d, e, k ) is the number of (necessarily cyclic) subgroups of order d in U ( e ). Each such group is the product of its Sylow p -subgroups forthe primes p dividing d . For each such p , the p -rank of U ( e ) is v ( p, e ),so U ( p ) contains ( p v ( p,e ) − / ( p −
1) subgroups of order p . Taking theproduct over p gives the number of subgroups of order d . Summingover d yields the formula (1) for the number of isomorphism types ofgroups of order n . Proposition 3.5.
Let G = G ( d, e, k ) be a group of squarefree order n as in Lemma 3.2. Let z = gcd( e, k − and g = e/z , so that we havefactorisations e = gz and n = de = dgz . Then the centre Z ( G ) of G is the cyclic group h σ g i of order z , and the commutator subgroup G ′ of G is the the cyclic group h σ z i of order g .Proof. For γ = σ a τ b ∈ G , we have σ − γσ = σ a − k b τ b . Since ord e ( k ) = d , it follows that γ commutes with σ if and only if d | b . But then ALI A. ALABDALI AND NIGEL P. BYOTT γ = σ a and τ γτ − = τ σ a τ − = σ ak . Thus τ γτ − = γ precisely when e | a ( k − g | a . Hence Z ( G ) = h σ g i .Turning to G ′ , we have τ στ − σ − = σ k − . Thus G ′ contains thenormal subgroup h σ k − i = h σ z i of G . Equality holds since G/ h σ k − i isabelian. (cid:3) We next find the number of isomorphism classes of groups G corre-sponding to the factorisation n = dgz . Proposition 3.6.
Let n = dgz be squarefree. Then the number ofisomorphism types of groups G of order n with | Z ( G ) | = z and | G ′ | = g is (3) ϕ ( d ) − X f | g µ (cid:18) gf (cid:19) Y p | d ( p v ( p,f ) − . Proof.
We keep d and e = n/d fixed. For each factor g of e let m ( g )be the number of isomorphism types of groups G = G ( d, e, k ) (with k varying) for which | G ′ | = g . We need to show that m ( g ) is given bythe formula (3).Let m ∗ ( g ) be the number of groups G ( d, e, k ) for which | G ′ | divides g . Then m ∗ ( g ) = X f | g m ( f ) , and so, by M¨obius inversion,(4) m ( g ) = X f | g m ∗ ( f ) µ (cid:18) gf (cid:19) . The distinct isomorphism types of groups G correspond to distinctsubgroups D of order d in U ( e ) ∼ = Q q | e U ( q ), where the product isover primes q dividing e . Let f | e . Then | G ′ | divides f preciselywhen e/f divides | Z ( G ) | , which occurs when D has trivial projectionin the factor U ( q ) for each prime q dividing e/f . Hence m ∗ ( f ) is thenumber of subgroups of order d in U ( f ), and, arguing as in Remark3.4, this is Q p | d ( p v ( p,f ) − / ( p − Q p | d ( p −
1) = ϕ ( d ), we obtain the expression (3) for m ( g ). (cid:3) Automorphisms and the Holomorph
For this section and the next, we fix a group G = G ( d, e, k ) of square-free order n . We keep the preceding notation, so g = | G ′ | , z = | Z ( G ) | ,and n = de = dgz . Our goal is to find the number of cyclic subgroupsof Hol( G ) which are regular on G . By Lemma 2.1, this will enableus to find the number of Hopf-Galois structures of type G on a cyclic OUNTING HOPF-GALOIS STRUCTURES 7 extension of degree n . In this section, we will describe Aut( G ) andHol( G ). In §
5, we determine all regular cyclic subgroups in Hol( G )and complete the proof of Theorem 1. In §
6, we sum over the differentisomorphism types G to prove Theorem 2.Until the end of §
6, we shall systematically use the notation p forprime factors of d and q for prime factors of e . Thus the primes q areof two types: either q | g or q | z .We begin by recording a formula which allows us to perform calcu-lations in G itself. For integers h and j ≥
0, we define(5) S ( h, j ) = j − X i =0 h i . In particular, S ( h,
0) = 0. A simple induction shows that, for any a ∈ Z ,(6) ( σ a τ ) j = σ aS ( k,j ) τ j . The next result describes the automorphisms of G . Lemma 4.1.
We have | Aut( G ) | = gϕ ( e ) and Aut( G ) ∼ = C g ⋊ U ( e ) , where a ∈ U ( e ) acts on C g by x x a . (Note that in general this actionis not faithful.)Explicitly, Aut( G ) is generated by the automorphism θ and automor-phisms φ s for each s ∈ U ( e ) , where (7) θ ( σ ) = σ, θ ( τ ) = σ z τ, and (8) φ s ( σ ) = σ s , φ s ( τ ) = τ. These automorphisms satisfy the relations (9) θ g = id G , φ s φ t = φ st , φ s θφ − s = θ s . Proof.
We first verify the existence of the automorphisms θ and φ s .Since ( σ z τ ) σ ( σ z τ ) − = σ k , (7) will determine an automorphism θ pro-vided that σ z τ has order d . By (6), this will hold if e | zS ( k, d ), thatis, if g | S ( k, d ). But for each prime q | g , we have k d ≡ k (mod q ),so that S ( k, d ) = k d − k − ≡ q ) . Thus g | S ( k, d ), as required. This shows the existence of the automor-phism θ . For s ∈ U ( e ), the element σ s has order e and τ σ s τ − = ( σ s ) k .It follows that there is an automorphism φ s as given in (8). ALI A. ALABDALI AND NIGEL P. BYOTT
It is clear that θ has order g . The remaining relations in (9) areeasily verified by checking them on the generators σ , τ of G .We have now shown that θ and the φ s generate a subgroup of Aut( G )isomorphic to C g ⋊ U ( e ). This subgroup has order gϕ ( e ). It remainsto show that there are no further automorphisms.Let ψ ∈ Aut( G ). As h σ i is a characteristic subgroup of G , being theunique subgroup of order e , we have ψ ( σ ) = σ s for some s ∈ U ( e ). Let ψ ( τ ) = σ a τ b with 0 ≤ b < d . Since ψ must satisfy ψ ( τ ) ψ ( σ ) ψ ( τ ) − = ψ ( σ ) k , we have σ sk b = σ sk and hence b = 1. Thus, by (6) again, ψ ( τ ) d = σ aS ( k,d ) , so that aS ( k, d ) ≡ e ). In particular, for each prime q dividing z , we have q | aS ( k, d ). But S ( k, d ) ≡ d q ) since k ≡ q ). Thus q | a . It follows that a = zc for some c ∈ Z , so ψ = θ c φ s . (cid:3) We now consider the holomorph Hol( G ) = G ⋊ Aut( G ) of G . Wewrite an element of this group as [ α, ψ ], where α ∈ G and ψ ∈ Aut( G ).The multiplication in Hol( G ) is given by(10) [ α, ψ ][ α ′ , ψ ′ ] = [ αψ ( α ′ ) , ψψ ′ ] . (The subgroup G in Hol( G ) is therefore identified with the left trans-lations in Perm( G ).) In view of Lemma 4.1, an arbitrary x ∈ Hol( G )can be written x = [ σ a τ b , θ c φ s ], where a ∈ Z e , s ∈ U ( e ), b ∈ Z d and c ∈ Z g . In Lemma 4.2 below, we will give a formula for powers of x inthe special case b = 1. We will then show in Proposition 4.3 why thiscase is all we need. We first introduce some further notation.Define(11) T ( k, s, j ) = j − X h =0 S ( s, h ) k h − for j ≥ , T ( k, s,
0) = 0 , where S ( s, h ) is given by (5). Note that we then have T ( k, s,
1) = 0and T ( k, s, j + 1) = T ( k, s, j ) + k j − S ( s, j ) for j ≥ . Lemma 4.2.
Let x = [ σ a τ, θ c φ s ] . Then, for j ≥ , we have (12) x j = [ σ A ( j ) τ j , θ cS ( s,j ) φ s j ] where A ( j ) = aS ( sk, j ) + czkT ( k, s, j ) .Proof. We argue by induction on j . When j = 0, we have S ( s,
0) = T ( k, s,
0) = 0 and A (0) = 0, so (12) holds in this case. Assuming (12) OUNTING HOPF-GALOIS STRUCTURES 9 for j , we have from (10) that x j +1 = [ σ A ( j ) τ j , θ cS ( s,j ) φ s j ][ σ a τ, θ c φ s ]= [ σ A ( j ) τ j ( θ cS ( s,j ) φ s j ( σ a τ )) , θ cS ( s,j ) φ s j θ c φ s ] . Thus, using (9), the second component of x j +1 is θ cS ( s,j ) φ s j θ c φ s = θ cS ( s,j ) θ cs j φ s j φ s = θ cS ( s,j +1) φ s j +1 , as required for (12). As for the first component of x j +1 , since θ cS ( s,j ) φ s j ( σ a τ ) = σ as j σ czS ( s,j ) τ, we have σ A ( j ) τ j ( θ cS ( s,j ) φ s j ( σ a τ )) = σ A ( j ) τ j σ as j σ czS ( s,j ) τ = σ A ( j ) σ as j k j σ czS ( s,j ) k j τ j +1 . We write this as σ A ′ τ j +1 , and calculate A ′ = A ( j ) + as j k j + czS ( s, j ) k j = a ( S ( sk, j ) + ( sk ) j ) + czk [ T ( k, s, j ) + k j − S ( s, j )]= aS ( sk, j + 1) + czkT ( k, s, j + 1)= A ( j + 1) . Thus (12) holds with j replaced by j + 1. This completes the induction. (cid:3) Proposition 4.3.
Let C be a cyclic subgroup of Hol( G ) which is regularon G . Then C is generated by some element x = [ σ a τ, θ c φ s ] , in which τ occurs with exponent . In fact, C contains precisely ϕ ( e ) generators of this type.Proof. For any ψ ∈ Aut( G ) and arbitrary α = σ a τ b ∈ G , we have ψ ( α ) = σ a ′ τ b for some a ′ ∈ Z . This is clear from Lemma 4.1 as itholds for ψ = φ s and ψ = θ . It then follows from (10) that thefunction Hol( G ) → h τ i , given by [ σ a τ b , ψ ] τ b , is a group homomor-phism. (This is not automatic, since the function Hol( G ) → G given by[ σ a τ b , ψ ] σ a τ b , is not in general a homomorphism.) In particular, forany x = [ σ a τ b , θ c φ s ] ∈ Hol( G ) and any j ≥
1, we have x j = [ σ A τ bj , ψ ]for some A ∈ Z e and some ψ ∈ Aut( G ), both depending on j . Thepermutation x j of G takes 1 G to σ A τ bj .Now let C be a regular cyclic subgroup of Hol( G ), and let x =[ σ a τ b , θ c φ s ] be a generator. Thus x has order n . Since C is transi-tive on G , the elements σ A τ bj , as j varies, must run through all el-ements of G . In particular, bj must run through all residue classes modulo d . Hence gcd( b, d ) = 1, and there exists f ≥ bf ≡ d ). Since gcd( d, e ) = 1, we may further assume that gcd( f, e ) =1. Then gcd( f, n ) = 1, so that x f is also a generator of C , and x f = [ σ A ′ τ bf , ψ ′ ] = [ σ A ′ τ, ψ ′ ] for some A ′ and ψ ′ . Replacing x by x f , we have found a generator of C with b = 1, as required.Now let x be any such generator. Then x j will be another if and onlyif gcd( j, n ) = 1 and j ≡ d ). The number of such generators istherefore ϕ ( n ) /ϕ ( d ) = ϕ ( e ). (cid:3) Hopf-Galois structures of type G As a first step towards determining when the element x in Proposi-tion 4.3 generates a regular subgroup, we give a condition for transi-tivity. Lemma 5.1.
Let x = [ σ a τ, θ c φ s ] ∈ Hol( G ) . Then the subgroup h x i of Hol( G ) acts transitively on G if and only if h x d i acts transitively on h σ i .Proof. Let h x i be transitive on G . Then, for each i ∈ Z , there is some j such x j · G = σ i . Then d | j by (12). Thus h x d i acts transitively on h σ i .Conversely, suppose that h x d i acts transitively on h σ i . Let σ i τ j ∈ G .By Lemma 4.2, we have x − j · σ i τ j ∈ h σ i . As h x d i is transitive on h σ i ,there is some h ∈ Z with x dh − j · σ i τ j = 1 G . Thus the arbitrary element σ i τ j lies in the same orbit under h x i as 1 G , so that h x i is transitive on G . (cid:3) In order to study the orbits of h x d i on h σ i , we examine the congruenceproperties of the sums S ( k, j ) and T ( k, s, j ) defined in (5) and (11)when j is a multiple of d . Proposition 5.2.
Let q be a prime dividing e . In the following, all con-gruences are modulo q . We will omit the modulus for brevity. Abusingnotation, we will write uv in such a congruence to denote uv ∗ where vv ∗ ≡ . (This notation will only be used when v .) (i) For any s , i ∈ Z with i ≥ , we have S ( s, di ) ≡ di if s ≡ s di − s − otherwise . OUNTING HOPF-GALOIS STRUCTURES 11 (ii)
Recall that k d ≡ . If also k then, for any s , i ∈ Z with i ≥ , we have T ( k, s, di ) ≡ dik ( k − if s ≡ dik ( s − if sk ≡ s di − k ( s − sk − otherwise . Proof. (i) This is immediate.(ii) The case i = 0 is clear, so assume i ≥
1. First let s ≡
1. Then S ( s, j ) ≡ j , so( k − T ( k, s, di ) = di − X j =0 ( k − jk j − = di − X j =0 jk j − di − X j =1 jk j − = di − X j =0 jk j − di − X j =0 ( j + 1) k j = di − X j =0 jk j − di − X j =0 ( j + 1) k j + dik di − = − di − X j =0 k j + dik di − . As k d ≡ k , we then have( k − T ( k, s, di ) ≡ dik di − , giving the result for s ≡ s T ( k, s, di ) ≡ di − X j =0 (cid:18) s j − s − (cid:19) k j − ≡ k ( s − " di − X j =0 ( sk ) j − di − X j =0 k j . The second sum vanishes mod q . The first is congruent to di if sk ≡ sk s then T ( k, s, di ) ≡ k ( s − di − X j =0 ( sk ) j ≡ k ( s − sk − (cid:0) ( sk ) di − (cid:1) ≡ k ( s − sk − (cid:0) s di − (cid:1) . (cid:3) Lemma 5.3.
Let x = [ σ a τ, θ c φ s ] ∈ Hol( G ) , so a ∈ Z e , c ∈ Z g , s ∈ U ( e ) . Then x generates a regular cyclic subgroup of Hol( G ) if and onlyif the triple ( s, a, c ) satisfies the following conditions: (i) for each prime q | z , we have s ≡ q ) and q ∤ a ; (ii) for each prime q | g , either s ≡ q ) and c q ) , or s ≡ k − (mod q ) and ( s − a + cz q ) . Proof.
Suppose that h x i is regular, and hence transitive, on G . ByLemma 5.1, h x d i is transitive on h σ i . It follows using Lemma 4.2 thatthe expression A ( di ) = aS ( sk, di ) + czkT ( k, s, di )represents all residue classes mod e as i varies. In particular, A ( di )represents every residue class mod q for each prime factor q of e . Weinvestigate this condition for each q in turn. Again, we omit the mod-ulus in congruences modulo q .First, let q | z , so k ≡
1. If s
1, then sk
1, so by Proposition5.2(i) we have A ( di ) ≡ a ( s di − s − , which cannot represent all residue classes mod q since there is no i suchthat s di ≡
0. On the other hand, if s ≡ A ( di ) ≡ adi. Since q ∤ d , this represents all residue classes mod q precisely when q ∤ a . Thus (i) holds. OUNTING HOPF-GALOIS STRUCTURES 13
Now let q | g , so k k d ≡
1. If s s k − , then, usingboth parts of Proposition 5.2, we have A ( di ) ≡ (cid:18) ( sk ) di − sk − (cid:19) a + czk (cid:18) s di − k ( s − sk − (cid:19) = ( s di − s − sk −
1) (( s − a + cz ) . Again, this cannot represent all residue classes mod q since s di s ≡ k and s ≡ k −
1. If s ≡ k then, as ( sk ) d ≡ k d ≡
1, we have(14) A ( di ) ≡ czk (cid:18) dik ( k − (cid:19) = czdik − . As q ∤ zd , this represents all residue classes mod q precisely when q ∤ c ,giving the first case in (ii).If s ≡ k − A ( di ) ≡ adi + czk (cid:18) dik ( s − (cid:19) ≡ dis − s − a + cz ) . (15)This represents all residue classes mod q precisely when ( s − a + cz
0, giving the second case in (ii).We have now shown that if x generates a regular cyclic subgroup,then (i) and (ii) hold.Conversely, suppose that (i) and (ii) hold. Then, by Proposition 5.2,the congruences (13), (14) and (15) hold modulo all relevant q . Foreach prime q | e , we then have that A ( di ) represents all residue classesmod q as i runs through any complete set of residues mod q . By theChinese Remainder Theorem, A ( di ) then ranges through all residueclasses mod e as i does. By Lemma 4.2, h x d i is then transitive on h σ i ,so h x i is transitive on G by Lemma 5.1. Finally, (13), (14) and (15)show that A ( de ) ≡ e ), so that x n = 1 G . Hence h x i is regularon G . (cid:3) Proof of Theorem 1.
By Proposition 4.3, any regular cyclic subgroupof Hol( G ) is generated by an element x as in Lemma 5.3. We countthe number of triples ( s, a, c ) satisfying the conditions there. As thereis only one possibility for s mod q when q | z but two when q | g ,there are 2 ω ( g ) possibilities for s (mod e ). Let us fix s and considerthe possibilities for a and c . For each prime q | z , condition (i) in Lemma 5.3 excludes one possibility for a mod q . For each q | g , wemay choose a mod q arbitrarily, and then, in either case of condition(ii), one possibility for c mod q is excluded. Thus we have ϕ ( z ) g choicesfor a mod e , and then ϕ ( g ) choices for c mod g . The number of ele-ments x = [ σ a τ, θ c φ s ] which generate a regular subgroup is therefore2 ω ( g ) ϕ ( z ) gϕ ( g ). By Proposition 4.3, each regular cyclic subgroup con-tains ϕ ( e ) = ϕ ( z ) ϕ ( g ) such generators, so there are 2 ω ( g ) g of thesesubgroups. Thus, using Lemma 2.1 and Lemma 4.1 (and writing C n for the cyclic group of order n ), we find that the number of Hopf-Galoisstructures of type G is | Aut( C n ) || Aut( G ) | ω ( g ) g = ϕ ( n ) gϕ ( e ) 2 ω ( g ) g = 2 ω ( g ) ϕ ( d ) . (cid:3) Proof of Theorem 2
In this section, we obtain the total number of Hopf-Galois structureson a cyclic field extension of squarefree degree n , thereby completingthe proof of Theorem 2.For each factorisation n = dgz , we have seen in Proposition 3.6 thatthe number of corresponding isomorphism types of group G is ϕ ( d ) − X f | g µ (cid:18) gf (cid:19) Y p | d ( p v ( p,f ) − . We have also seen in Theorem 1 that there 2 ω ( g ) ϕ ( d ) Hopf-Galois struc-tures of each of these types. To obtain the total number of Hopf-Galoisstructures, we simply multiply these two quantities and sum over fac-torisations of n . This yields X dgz = n ω ( g ) ϕ ( d ) ϕ ( d ) − X f | g µ (cid:18) gf (cid:19) Y p | d ( p v ( p,f ) − . Setting t = g/f and noting that ω ( g ) = ω ( t ) + ω ( f ), we can rewritethe previous sum as X dftz = n µ ( t )2 ω ( t ) ω ( f ) Y p | d ( p v ( p,f ) − . Let m = tz , and observe that µ ( t ) = ( − ω ( t ) . The sum then becomes X dfm = n X t | m ( − ω ( t ) ω ( f ) Y p | d ( p v ( p,f ) − . OUNTING HOPF-GALOIS STRUCTURES 15
Recall that a function F on the natural numbers is said to be multi-plicative if F ( rs ) = F ( r ) F ( s ) whenever gcd( r, s ) = 1. The function t ( − ω ( t ) is clearly multiplicative, and hence so is the function m P t | m ( − ω ( t ) . However, evaluating this last function at a prime q gives ( − ω (1) + ( − ω ( q ) = 1 + ( −
2) = − µ ( q ). As µ is alsomultiplicative, it follows that P t | m ( − ω ( t ) = µ ( m ) for squarefree m .(This is not true for arbitrary natural numbers m .) Hence the totalnumber of Hopf-Galois structures on a cyclic extension of squarefreedegree n is X dfm = n µ ( m )2 ω ( f ) Y p | d ( p v ( p,f ) − . After a change of notation, this gives the formula (2), completing theproof of Theorem 2. 7.
Examples
In this section, we give some examples and show how several re-sults in the literature can be obtained as special cases of our results.Throughout, n is a squarefree integer and L/K is a cyclic Galois ex-tension of degree n .7.1. Cyclic Hopf-Galois Structures.
The group G ( d, e, k ) in Lemma3.2 is cyclic only when d = 1, e = n . Indeed, this is the only case where G ( d, e, k ) is abelian, or even nilpotent. In this case z = n , g = 1, andTheorem 1 shows that there is only one Hopf-Galois structure of cyclic(or abelian, or nilpotent) type on L/K . This can also be seen from[Byo13, Theorem 2]. The unique cyclic Hopf-Galois structure is ofcourse the classical one.When gcd( n, ϕ ( n )) = 1, there are no other groups G ( d, e, k ), andhence there are no Hopf-Galois structures on L/K beyond the classicalone. This was shown, together with its converse, in [Byo96]. The casethat n is prime occurs in [Chi89].7.2. Dihedral Hopf-Galois Structures.
Let n = 2 m where m is anodd squarefree number. The group G ( d, e, k ) in Lemma 3.2 is dihedralwhen d = 2 and k = − ∈ U ( e ). Then e = g = m . It follows fromTheorem 1 that a cyclic extension of degree n admits 2 ω ( m ) Hopf-Galoisstructures of dihedral type.7.3.
Two Primes.
Let n = pq for primes p > q . We assume that q | ( p −
1) (since otherwise the only group of order n is the cyclicgroup C n ). Up to isomorphism, there are two groups of order n , thecyclic group C n (with d = 1, g = 1, e = z = pq ) and the metabelian d g z Term in (2)1 pq p q − q p −
21 1 pq q p q − Table 1.
Nonzero terms in (2) for n = pq .group M = C p ⋊ C q where C q acts nontrivially on C p (so d = q , e = g = p , z = 1). As we have seen in § n admits just one Hopf-Galois structure of cyclic type (namely theclassical one). By Theorem 1, it also admits 2 ω ( p ) ϕ ( q ) = 2( q −
1) Hopf-Galois structures of type M . This result was obtained in [Byo04b],where we also considered Hopf-Galois structures on a Galois extensionwith Galois group M . When q = 2, the result follows from § n = dgz , and thatTheorem 2 correctly counts the total number of Hopf-Galois structures,in this case.In the sum (3) of Proposition 3.6, Q p | d ( p v ( p,f ) −
1) vanishes unless d = 1 or d = q , f = p (so that also g = p ). When d = 1, (3) reduces to X f | g µ (cid:18) gf (cid:19) = ( g = 1;0 otherwise.Thus when d = 1, to get a group G ( d, e, k ) we must take g = 1 and z = pq . We then have G ( d, e, k ) = C n . When d = 1, all terms in (3)vanish unless d = q , g = p , when the term for f = p gives ϕ ( q ) − ( q −
1) = 1. Thus (3) tells us that there is just one isomorphism type ofnonabelian group of order n . Hence Proposition 3.6 does indeed givethe correct number of isomorphism types for each factorisation. Bysimilar reasoning (which we leave to the reader), H¨older’s formula (1)correctly predicts two isomorphism classes of groups of order n .We now turn to Theorem 2. The product over p | d vanishes unless d = 1 or d = q , g = p . The nonzero contributions to (2) for thevarious factorisations n = gzd are shown in Table 1. Summing thefinal column of Table 1 gives the correct count of 2( q −
1) + 1 Hopf-Galois structures on a cyclic extension of degree n = pq . Table 1also illustrates an important feature of the formula (2): factorisations OUNTING HOPF-GALOIS STRUCTURES 17 n = dgz for which there are no corresponding groups G can neverthelesscontribute nonzero terms to (2).7.4. Three Primes.
Let n = p p p where p < p < p are primes.Both the number of isomorphism classes of groups of order n , andthe number of Hopf-Galois structures on a cyclic extension of degree n , depend on congruence conditions relating the three primes. Thereare two combinations of these conditions for which the Hopf-Galoisstructures on all Galois extensions of degree p p p (not just cyclicextensions) have been enumerated.The first of these is when p = 2 and p = 2 p + 1 (so p is a SophieGermain prime and p is a safeprime). Kohl [Koh13, Theorem 5.1]treated this case as an application of his method for studying Hopf-Galois structures on Galois extensions of degree mp (with p prime and m < p ). Those extensions with Galois group Hol( C p ) = C p ⋊ C p − had previously been considered in [Chi03].The second situation where all Hopf-Galois structures have been de-termined is when p > p ≡ p ≡ p ) but p p ). This case is treated in [Koh16, Theorem 2.4]. The sametechniques could be applied to other combinations of congruence con-ditions, but separate calculations would be required for each case.In the following, we will apply Theorem 1 to count the Hopf-Galoisstructures only on a cyclic extension of degree n = p p p , but under allpossible combinations of congruence conditions. In particular, this willrecover those parts of Kohl’s results in [Koh13, Koh16] which relate tocyclic extensions.In Table 2 we show the factorisations n = dgz for which groups exist,the number of isomorphism types of these groups, and the number ofHopf-Galois structures of each isomorphism type.The first column of Table 2 numbers the factorisations for ease ofreference, and the factorisation is shown in the next 3 columns. The5th column shows the congruence conditions which must be satisfiedfor groups to exist. The 6th column shows the number of isomorphismtypes of group corresponding to the given factorisation, as given byProposition 3.6. These can also be found directly, as explained below.The final column shows the number of Hopf-Galois structures for eachisomorphism type. This is given by the formula 2 ω ( g ) ϕ ( d ) of Theorem1. We now explain how to find the values in the 6th column of Table2 directly. (This illustrates in simple cases the proof of Proposition3.6.) Consider for example case 2, where d = p , g = p , z = p ,so e = p p . The distinct isomorphism types of groups G ( d, e, k ) with Case d g z
Condition p p p p p p p ≡ p ) 1 2( p − p p p p ≡ p ) 1 2( p − p p p p ≡ p ≡ p ) p − p − p p p p ≡ p ) 1 2( p − p p p p ≡ p p ) 1 2( p − p − Table 2.
Numbers of isomorphism types and Hopf-Galois structures for n = p p p .these parameters correspond to subgroups h k i ⊆ U ( p p ) of order p forwhich z = gcd( k − , p p ) = p . Since k ≡ p ), we can identify h k i with a subgroup of order p in U ( p ). Such a subgroup exists since p ≡ p ), and it is unique since U ( p ) is cyclic. Thus there isjust one group G ( d, e, k ) in case 2. In case 4, however, where d = p , g = p p and z = 1, the isomorphism types of groups G ( d, e, k ) correspondto subgroups h k i ⊆ U ( p p ) of order p with gcd( k − , p p ) = 1. Now U ( p p ) ∼ = U ( p ) × U ( p ) contains p + 1 subgroups of order p . Forone of these, gcd( k − , p p ) = p . This gives the group G just foundin case 2. Another of the subgroups has gcd( k − , p p ) = p , andthis is counted in case 3. The remaining p − U ( p p )give groups G with g = p p and z = 1. Thus the number of groupsrecorded in case 4 is p − n , treating each combination of relevant congruenceconditions on p , p , p separately. The results are shown in Table 3.For each combination of congruence conditions, we pick out the casesfrom Table 2 where any groups G ( d, e, k ) exist. To obtain the totalnumber of isomorphism types of groups of order n , we add the numbersof groups from the corresponding rows in Table 2, giving the entries inthe 5th column of Table 3. These agree with the values given by Kohl[Koh16, p. 46]. To obtain the total number of Hopf-Galois structures,we multiply the entries in the final two columns of Table 2 and addthese values for the appropriate rows. After simplification, this givesthe entries in the final column of Table 3.We now specialise to the two situations considered in [Koh13, Theo-rem 5.1] and [Koh16, Theorem 2.4] in order to confirm that we recoverthose parts of Kohl’s results pertaining to cyclic extensions.First let p = 2 and let p = 2 p + 1 be a safeprime. Thus wehave p j ≡ p i ) whenever 1 ≤ i ≤ j ≤
3, corresponding to the
OUNTING HOPF-GALOIS STRUCTURES 19 p | ( p − p | ( p − p | ( p −
1) Cases p − p − p + 2 (2 p − yes no no 1, 5 2 2 p − p + 2 p − p p − p + 4 4 p + 2 p p − p + 1 Table 3.
Total numbers of Hopf-Galois structures for n = p p p .final row (“yes–yes–yes”) in our Table 3. The first row of the tablein [Koh13, Theorem 5.1] shows that there are 6 isomorphism types ofgroups of order n = p p p , which Kohl denotes by C mp , C p × D q , F × C , C q × D p , D pq , Hol( C p ), where q = p and p = p . These contribute 1,2, 2( p − p −
1) Hopf-Galois structures respectively. Thetotal number of Hopf-Galois structures is therefore 4 p + 5. Thesegroups are respectively those of cases 1, 2, 5, 3, 4, 6 in our Table 2.Putting p = 2 in Table 2, we again get 4 p + 5 for the total numberof Hopf-Galois structures, and the number of Hopf-Galois structuresof each type shown in Table 2 agrees with Kohl’s values. Thus ourresults recover the part of Kohl’s result [Koh13, Theorem 5.1] relatingto cyclic extensions of degree 2 pq = 2 p (2 p + 1).Now let p > p ≡ p ≡ p ) but p p ),corresponding to the 4th row (“no–yes–yes”) of our Table 3. The firstrow of the table in [Koh16, Theorem 2.4] shows that there are p + 2groups G . (Note that the final column, headed C p p ⋊ i C p , correspondsto p − ≤ i ≤ p − p − p − p − Hopf-Galois structures. This agrees with our count in Table 3 and therelevant cases, 1–4, in Table 2. (The restriction p > Four primes.
As a final example, we consider the case when n = p p p p is the product of 4 distinct primes, under the assumptionthat(16) p j ≡ p i ) whenever i < j. d g z p p p p p p p p p − p p p p p − p p p p p − p p p p p − p − p p p p p − p − p p p p p − p − p p p p p − p − p p p p p − p p p p p − p p p p p − p − p p p p p − p p p p p − p − p p p p p − p − p p p p p + 1)( p + 1) − p − p − p p p p p − p − p p p p p − p − p p p p p − p − p − Table 4.
Numbers of isomorphism types and Hopf-Galois structures for n = p p p p .(Thus we have p < p < p < p .)We record in Table 4 the number of isomorphism classes of groups G ( d, e, k ), and the number of Hopf-Galois structures of each type, cor-responding to each relevant factorisation n = dgz .It follows from this table that, under the assumption (16), thereare p + p p + 2 p + 2 p + 8 isomorphism types of groups of order n = p p p p , and the total number of Hopf-Galois structures is4 p p + 8 p + 2 p p p − p − p p + 10 p − . For example, if n = 2 · · ·
43 = 1806, or more generally, if n = 42 p forany prime p ≡ n admitsprecisely 211 Hopf-Galois structures of 28 different types.When (16) does not hold, we can enumerate the Hopf-Galois struc-tures by picking out the appropriate rows in Table 4, just as we did in § OUNTING HOPF-GALOIS STRUCTURES 21
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E-mail address : [email protected] (N. Byott) Department of Mathematics, College of Engineering,Mathematics and Physical Sciences, University of Exeter, Exeter EX44QF U.K. E-mail address ::